Download List of axioms and theorems of Incidence geometry

Axioms of Incidence Geometry
Incidence Axiom 1. There exist at least three distinct noncollinear points.
Incidence Axiom 2. Given any two distinct points, there is at least one line that contains both of them.
Incidence Axiom 3. Given any two distinct points, there is at most one line that contains both of them.
Incidence Axiom 4. Given any line, there are at least two distinct points that lie on it.
Parallel Postulates
The Elliptic Parallel Postulate. For each line ` and each point A that does not lie on `, there are no lines that contain A and are parallel to `.
The Euclidean Parallel Postulate. For each line ` and each point A that does not lie on `, there is one and only one line that contains A and
is parallel to `.
The Hyperbolic Parallel Postulate. For each line ` and each point A that does not lie on `, there are at least two distinct lines that contain
A and are parallel to `.
Theorems of Incidence Geometry
Theorem 2.25. Given any point A, there exists another point that is distinct from A.
Theorem 2.26. Given any point, there exists a line that contains it.
Corollary 2.27. If A and B are points (not necessarily distinct), there is a line that contains both of them.
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Theorem 2.28. If ` is a line and A and B are two distinct points on `, then AB D `.
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Theorem 2.29. If A and B are distinct points, and C is any point that does not lie on AB , then A, B, and C are noncollinear.
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Theorem 2.30. If A, B, and C are noncollinear points, then A and B are distinct, and C does not lie on AB .
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Corollary 2.31. If A, B, and C are noncollinear points, then A, B, and C are all distinct. Moreover, A does not lie on BC , B does not lie
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on AC , and C does not lie on AB .
Theorem 2.32. Given a line ` and a point A that lies on `, there exists a point B that lies on ` and is distinct from A.
Theorem 2.33. Given any line, there exists a point that does not lie on it.
Theorem 2.34. Given two distinct points A and B, there exists a point C such that A, B, and C are noncollinear.
Theorem 2.35. Given any point A, there exist points B and C such that A, B, and C are noncollinear.
Theorem 2.36. Given two distinct points A and B, there exists a line that contains A but not B.
Theorem 2.37. Given any point, there exists a line that does not contain it.
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Theorem 2.38. If A, B, and C are noncollinear points, then AB ¤ AC .
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Corollary 2.39. If A, B, and C are noncollinear points, then AB , AC , and BC are all distinct.
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Theorem 2.40. If A, B, and C are collinear points, and neither B nor C is equal to A, then AB D AC .
Theorem 2.41. Given two distinct, nonparallel lines, there exists a unique point that lies on both of them.
Theorem 2.42. Given any point, there are at least two distinct lines that contain it.